Theorem scafvalg 13402

Description: The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
scaffval.b	|- B = ( Base ` W )
scaffval.f	|- F = (Scalar ` W )
scaffval.k	|- K = ( Base ` F )
scaffval.a	|- .xb = ( .sf ` W )
scaffval.s	|- .x. = ( .s ` W )

Assertion

Ref	Expression
scafvalg	|- ( ( W e. V /\ X e. K /\ Y e. B ) -> ( X .xb Y ) = ( X .x. Y ) )

Proof of Theorem scafvalg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		scaffval.b	. . . 4 |- B = ( Base ` W )
2		scaffval.f	. . . 4 |- F = (Scalar ` W )
3		scaffval.k	. . . 4 |- K = ( Base ` F )
4		scaffval.a	. . . 4 |- .xb = ( .sf ` W )
5		scaffval.s	. . . 4 |- .x. = ( .s ` W )
6	1, 2, 3, 4, 5	scaffvalg 13401	. . 3 |- ( W e. V -> .xb = ( x e. K , y e. B |-> ( x .x. y ) ) )
7	6	3ad2ant1 1018	. 2 |- ( ( W e. V /\ X e. K /\ Y e. B ) -> .xb = ( x e. K , y e. B |-> ( x .x. y ) ) )
8		oveq12 5886	. . 3 |- ( ( x = X /\ y = Y ) -> ( x .x. y ) = ( X .x. Y ) )
9	8	adantl 277	. 2 |- ( ( ( W e. V /\ X e. K /\ Y e. B ) /\ ( x = X /\ y = Y ) ) -> ( x .x. y ) = ( X .x. Y ) )
10		simp2 998	. 2 |- ( ( W e. V /\ X e. K /\ Y e. B ) -> X e. K )
11		simp3 999	. 2 |- ( ( W e. V /\ X e. K /\ Y e. B ) -> Y e. B )
12		vscaslid 12623	. . . . . 6 |- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
13	12	slotex 12491	. . . . 5 |- ( W e. V -> ( .s ` W ) e. _V )
14	5, 13	eqeltrid 2264	. . . 4 |- ( W e. V -> .x. e. _V )
15	14	3ad2ant1 1018	. . 3 |- ( ( W e. V /\ X e. K /\ Y e. B ) -> .x. e. _V )
16		ovexg 5911	. . 3 |- ( ( X e. K /\ .x. e. _V /\ Y e. B ) -> ( X .x. Y ) e. _V )
17	10, 15, 11, 16	syl3anc 1238	. 2 |- ( ( W e. V /\ X e. K /\ Y e. B ) -> ( X .x. Y ) e. _V )
18	7, 9, 10, 11, 17	ovmpod 6004	1 |- ( ( W e. V /\ X e. K /\ Y e. B ) -> ( X .xb Y ) = ( X .x. Y ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   _Vcvv 2739   ` cfv 5218  (class class class)co 5877   e. cmpo 5879   Basecbs 12464  Scalarcsca 12541   .scvsca 12542   .sfcscaf 13383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-5 8983  df-6 8984  df-ndx 12467  df-slot 12468  df-base 12470  df-sca 12554  df-vsca 12555  df-scaf 13385

This theorem is referenced by:  lmodfopne  13421